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Doing HLM using SAS PROC MIXED Kaz www.estat.us Feb 2005

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My points (1) Easy to compare HLM and other models that are not HLM; thus, helpful. This is because PROC MIXED lets you run models that are not HLM. (2) Easy to understand what makes HLM HLM. In SAS, what is not essential to HLM is done outside PROC MIXED (e.g., centering)

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OLS vs. HLM in PROC MIXED. The difference is a RANDOM statement. OLS regression syntax PROC MIXED ; MODEL Y= X; Run; HLM syntax PROC MIXED; Model Y=X; random intercept X/ subject=school; Run; (1) OLS Y_jk = b0 + error_jk (2) HLM Level1: Y_jk=b0 + error_jk Level2: b0=g0 + error_k or Y_jk=b0 + error_jk + error_k

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Again, turning a simple linear model into HLM (1) PROC MIXED; Model Y=X W; Run; (2) PROC MIXED; Model Y=X W; random intercept X W/subject=GroupID; Run; (3) Random statement below reads: I request that the intercept, as well as the effects of X and W be Estimated for each subject which can be identified by GroupID.

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How to write SAS PROC MIXED syntax: Intuitive way (1) Write all the variable names at the model statement. model Y=X W; (2) Decide which variables effect you want to estimate by schools random intercept X W/subject=school;

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More careful way 1.Start from level-specific specification. e.g., level1:y=b0 + b1*X + error_ij level2: b0=g00 + g01*W + error_0j level2: b1=g10 + g11*W + error_1j 2.Insert level-2 equations into level-1 equations. 3.Write the variable names involved in model statement. 4.Find random components (written in Roman alphabets) RULE1: Put intercept in the random statement to accommodate higher level errors. RULE2: If the name of any variables sits right next to level-2 error with an asterisk (e.g., X*level-2 error), put those variable names in the random statement. (RULE3:No worry about residual. It is set by default.)

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Example 1 Anova Model Level1: Y_ij=b0j + Residual_ij Level2: b0j= g00 + U_0j Y= g00 + U_0j + Residual proc mixed ; class group; model Y= ; random intercept/subject=school; run; I said: RULE1: Put intercept in the random statement to accommodate higher level errors. RULE2: If the name of any variables sits right next to level-2 error with an asterisk (e.g., X*level-2 error), put those variable names in the random statement. ONLY RULE1 relevant in this model.

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Example 2 Slope as outcome models Level1: Y_ij=b0j + b1j*X + Residual_ij Level2: b0j= g00 + g01*W + U_0j Level2: b1j=g10+ g11*W + U_1j Y= g00 + g01*W + g10*X + g11*W*X + U_1j*X + U_0j + Residual proc mixed ; class group; model Y= W X W*X; random intercept X/subject=school; run; What were.. RULE1? RULE2?

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How to do substitution: Cheating using HLM software! PUSH MIXED button to get a little window like this.

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How to do substitution by hand Level1: Y_ij=b0j + b1j*X + Residual_ij Level2: b0j= g00 + t g01*W + U_0j Level2: b1j=gt10+ g11*W + U_1j 1. Insert higher level equations into the level-1 equation. Y=[g00 + g01*W + U_0j] + [g10+g11*W + U_1j]*X + Residual_ij 2. Take out the brackets --> Y=g00 + g01*W + U_0j + g10*X +g11*W*X + U_1j *X + Residual_ij 3. Notice which parts are structural part and which parts are random components. Y=g00 + g01*W + g10*X +g11*W*X + U_1j *X + U_0j + Residual_ij proc mixed ; model W X W*X; random intercept X /subject=school; run; What were rule1 and rule 2?

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Fixed Effects or Random Effects OLS regression is a fixed effect model PROC MIXED; Model Y=X; Run; OLS regression is a model with fixed effects. So in a way OLS is a special case of HLM. This is an awfully inflexible model that does not consider the existence of various sources of errors. HLM PROC MIXED; Model Y=X; random intercept X/ subject=groupID; Run; If a researcher thinks the effect of X (and the intercept) is different by groups, so we should treat these coefficients as random effects.

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Benefit 1 of using random effect Conceptual one Useful to think about Micro-Macro problems (1) Student: Math score=b0 + b1*parents education level + …. + error Country: b1=g00 + g01*SELECTION + error (2) Classroom: teacher perception of math ability of class =b0 + b1*average parents education level +b2*average math score +b3*noise + error Country: b1=g10 + g11*National Exam + error b2=g20 + g21*National Exam + error b3=g30 + g31*National Exam + error

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Benefit 2: Statistical benefit Statistical Benefits –In deriving a grand mean (re: the effect of X or an intercept) HLM does shrinkage. This pulls inaccurate group means towards the grand mean, so we can reduce the influence of outliners if their estimates are inaccurate (i.e., having large error variance and/or coming from a small number of observations within each group unit) Shrunk School mean=reliability*school mean where reliability is a function of N of observation in a group unit and variance. (R&B HLM book, p. 48) Quiz: 1) what happens to a school whose reliability is 1? 2) What happens if all schools are 1 on reliability? 3) What happens if all schools are.5 on reliability?

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Quick decision rule Random or fix Do I open the door at 11PM? –Literature –Theory –Exploratory analysis (let s see what happens.)

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Moore complicated: Two step decisions regarding random effects (I need your help in phrasing this.) Step 1: Effect different by school? Step 2: Random or Fixed? –Fixed: Use a series of dummy variables (in reality too tedious) –Random: Shrinkage applies and get a precision guided grand mean

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Example Student Engagement study using ESM by Uekawa, Borman, and Lee

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Engagement Level (Rasch model composite) When you were signaled the first time today, SD D A SA I was paying attention ……………………….. O O O O I did not feel like listening …………………… O O O O My motivation level was high ……………….. O O O O I was bored ………………………………….. O O O O I was enjoying class ………………………… O O O O I was focused more on class than anything else O O O O I wished the class would end soon …………. O O O O I was completely into class ………………… O O O O The MEANS Procedure Analysis Variable : engagement engagement N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 2316 0.1167889 10.0106694 -31.5283511 26.4164991 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

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3-level HLM Level1: Repeated Measures (10 beeps) Level2: Students (10 kids from a class) Level3: courses (34 courses, Monday to Friday)

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3-level HLM Libname here "C:\"; /*This is three level model*/ proc mixed data=here.esm covtest noclprint; class IDclass IDstudent; model engagement= /solution ddfm=kr; random intercept /sub=IDstudent(IDclass); random intercept /sub=IDclass ; run; Quiz: how can we make this a 2-level hlm?

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PROC MIXED statement proc mixed data=here.esm covtest noclprint; covtest does a test for covariance components (whether variances are significantly larger than zero.). The reason why you have to request such a simple thing is that COVTEST is not based on chi-square test that one would use for a test of variance. It uses instead t-test or something that is not really appropriate. Shockingly, SAS has not corrected this problem for a while. Anyways, because SAS feels bad about it, it does not want to make it into a default option, which is why you have to request this. Not many people know this and I myself could not believe this. So I guess that means that we cannot really believe in the result of COVTEST and must use it with caution. When there are lots of group units, use NOCLPRINT to suppress the printing of group names.

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CLASS statement class IDclass IDstudent Hisp; We throw in the variables that we want SAS to treat as categorical variables. Variables that are characters (e.g., city names) must be on this line (it won t run otherwise). Group IDs, such as IDclass in my example data, must be also in these lines; otherwise, it won t run. Variables that are numeric but dummy-coded (e.g., black=1 if black;else 0) don t have to be in this line, but the outputs will look easier if you do. One thing that is a pain in the neck with CLASS statement is that it chooses a reference category by alphabetical order. Whatever group in a classification variable that comes the last when alphabetically ordered will be used as a reference group. We can control this by data manipulation. For example, if gender=BOY or GIRL, then I tend to create a new variable to make it explicit that I get girl to be a reference group: If gender= Boy then gender2= (1) Boy ; If gender= Girl then gender2= (2) Girl ;

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MODEL statement model engagement= /solution ddfm=kr; ddfm= kr specifics the ways in which the degree of freedom is calculated. It seems most close to the degree of freedom option used by Bryk, Raudenbush, and Congdon s HLM program. Could be computationally very heavy if a model is complicated. ddfm=bw would run faster, though DF would be wrong.

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Random statement random intercept X/sub=IDstudent(IDclass); random intercept X/sub=IDclass ; We can estimate variance of slopes for categorical variables using group= option --- without necessarily making them into dummy variables. random intercept race /sub=IDclass group=race; (instead of random intercept black white hispanic/sub=IDclass; )

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Libname here "G:\SAS"; proc mixed data=here.esm covtest noclprint; weight precision_weight; class IDclass IDstudent; model engagement= /solution ddfm=kr; random intercept /sub=IDstudent(IDclass); random intercept /sub=IDclass ; run; The Mixed Procedure Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept IDstudent(IDclass) 23.3556 2.5061 9.32 <.0001 Intercept IDclass 5.3168 2.0947 2.54 0.0056 Residual 31.9932 1.0282 31.12 <.0001 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept -1.2315 0.5081 30.2 -2.42 0.0216 Engagement=b0 + residual + e_student + e_course MODEL 1 RQ: How does the engagement score varies by students and class?

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Libname here "G:\SAS"; proc mixed data=here.esm covtest noclprint; weight precision_weight; class IDclass IDstudent subject; model engagement= hisp /solution ddfm=kr; random intercept /sub=IDstudent(IDclass); random intercept hisp /sub=IDclass ; run; Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept -0.6287 0.5101 33.1 -1.23 0.2265 hisp -2.2113 1.0031 17.7 -2.20 0.0410 Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept IDstudent(IDclass) 22.4743 2.4712 9.09 <.0001 Intercept IDclass 3.6944 1.7791 2.08 0.0189 hisp IDclass 5.9656 5.3290 1.12 0.1315 Residual 31.9871 1.0275 31.13 <.0001 MODEL 2 RQ: What is the effect of being Hispanic students? And is the effect vary by class?

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MODEL 3 proc mixed data=here.esm covtest noclprint; weight precision_weight; class IDclass IDstudent subject; model engagement= hisp math hisp*math /solution ddfm=kr; random intercept /sub=IDstudent(IDclass); random intercept hisp /sub=IDclass ; run; Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept -0.3249 0.7061 34.2 -0.46 0.6484 hisp -4.1236 1.4562 14.5 -2.83 0.0129 math -0.6081 1.0108 33.7 -0.60 0.5515 hisp*math 3.3305 1.9233 15.2 1.73 0.1035 The Mixed Procedure Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept IDstudent(IDclass) 22.6987 2.5020 9.07 <.0001 Intercept IDclass 3.5328 1.7254 2.05 0.0203 hisp IDclass 4.2309 5.0161 0.84 0.1995 Residual 31.9761 1.0269 31.14 <.0001 Hispanic alienation phenomenon related to subject? Math versus Science

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MODEL 3 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept -0.3249 0.7061 34.2 -0.46 0.6484 hisp -4.1236 1.4562 14.5 -2.83 0.0129 math -0.6081 1.0108 33.7 -0.60 0.5515 hisp*math 3.3305 1.9233 15.2 1.73 0.1035 The Mixed Procedure Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept IDstudent(IDclass) 22.6987 2.5020 9.07 <.0001 A Intercept IDclass 3.5328 1.7254 2.05 0.0203 B hisp IDclass 4.2309 5.0161 0.84 0.1995 -> C Residual 31.9761 1.0269 31.14 <.0001 Level1:engagement=b0+ b1*Hispanic + residual Level2: b0=g00 + A Level2: b1=g10 Level3: g00=t_000 + t_100*Math + B Level3: g10=t_100 + t_101*Math + C Level1:engagement= t_000 + t_100*Math + t_100*Hispanic + t_101*Math*Hispanic + C*Hispanic + B + A + residual VERSUS

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Which is easy to understand? In HLM softwareIn SAS PROC MIXED Level-1 Intercept Disappears Level-2 Intercept Disappears Level-3 Intercept Intercept Level-1 Error Residual Level-2 Error Random effects Level-3 Error Random effects HLM way Level1:engagement=b0+ b1*Hispanic + residual Level2: b0=g00 + A Level2: b1=g10 Level3: g00=t_000 + t_100*Math + B Level3: g10=t_100 + t_101*Math + C PROC MIXED way Level1:engagement= t_000 + t_100*Math + t_100*Hispanic + t_101*Math*Hispanic + C*Hispanic + B + A + residual

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Why do we center variables? Level1:engagement= t_000 + t_100*Math + t_100*Hispanic + t_101*Math*Hispanic + C*Hispanic + B + A + residual Imagine we have to report to teachers their students average engagement score. We want to use B + t_000. To be clear about Meaning of t_000 part, we could center variables, if it makes sense.

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What about Centering? In SAS, we use PROC STANDARD to do centering and this is outside of PROC MIXED. When I learned this, I thought, I have done it before! because centering is similar to the concept of Z-scores. This is GROUP MEAN Centering Proc standard data=X mean=0; by GroupID; var X; Run; This is GRAND MEAN Centering proc standard data=X mean=0; var X; Run; By the way, just for your information, this is to create Z-scores proc standard data=X mean=0 STD=1; var X; Run; When you use SAS PROC MIXED, you notice Centering is not really a topic that is specific to HLM because it is done outside PROC MIXED.

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What does it mean to center dummy variable, like gender? 1.To adjust for gender composition. 2. -Without it, the intercept = either male or female -With it, the intercept is adjusted for gender composition. 3. See my Excel Presentation if we have time. www.estat.us/sas/centering.xls

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END To go back to my HLM page www.estat.us/id38.html

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